Group #34000656 — subgroups

Subgroups — #34000656

Order 18688.

190 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625491	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989449	C₂	1	1	yes	maximal, characteristic
3	#22611731	C₂	1	1	yes	maximal, characteristic
4	#17989608	—	2	1	no	—
5	C₂×C₄⋊#10596833	—	2	1	no	—
6	#17989451	—	2	1	no	—
7	#17989449	—	2	1	no	—
8	#17989449	C₄	1	2	yes	—
9	#17989448	C₄	1	2	yes	—
10	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
11	#17988405	—	2	1	no	—
12	C₂²⋊#11924003	C₄	1	1	yes	characteristic
13	#17988362	—	2	1	no	—
14	C₂²⋊C₁₆×C₇₃	C₄	1	1	yes	characteristic
15	C₂²×#10596833	—	4	1	no	—
16	C₄⋊#10596833	—	2	2	no	—
17	C₄⋊#10596833	—	2	2	no	—
18	C₄⋊#10596833	—	4	1	no	—
19	C₂²⋊#10596833	—	4	1	no	—
20	#14556646	—	4	1	no	—
21	C₂²×#10596831	Q₈	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
23	C₂×#11924003	—	2	1	no	—
24	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₇₃	D₄	1	1	yes	characteristic
27	C₈×Dic₇₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₃	—	2	1	no	—
30	C₂×#10596833	—	4	2	no	—
31	C₂×#10596833	—	4	2	no	—
32	C₂×#10596833	—	8	1	no	—
33	C₂×#10596833	—	8	1	no	—
34	C₂²×Dic₇₃	SD₁₆	1	1	yes	characteristic
35	C₂²×Dic₇₃	SD₁₆	1	1	yes	characteristic
36	C₂×#10596831	—	2	2	no	—
37	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
38	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
39	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₇₃	D₈	1	1	yes	characteristic
42	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
43	#11924003	—	4	1	no	—
44	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	4	1	no	—
46	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₁₆×C₇₃	—	4	1	no	—
49	C₂³×C₇₃	#106	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
54	C₂×Dic₇₃	—	2	1	no	—
55	C₂×Dic₇₃	—	2	1	no	—
56	C₂×Dic₇₃	—	4	1	no	—
57	C₂×Dic₇₃	#102	1	1	yes	characteristic
58	C₂×Dic₇₃	#102	1	1	yes	characteristic
59	C₂×Dic₇₃	#103	1	1	yes	characteristic
60	C₂×Dic₇₃	#103	1	1	yes	characteristic
61	#10596833	—	8	2	no	—
62	#10596833	—	8	2	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	—	2	1	no	—
65	#10596831	—	2	1	no	—
66	#10596831	#108	1	2	yes	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	#340	1	1	yes	characteristic
70	C₂²×C₇₃	#365	1	1	yes	characteristic
71	C₂²×C₇₃	#365	1	1	yes	characteristic
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	#341	1	1	yes	characteristic
74	C₄×C₇₃	#344	1	1	yes	characteristic
75	Dic₇₃	—	2	1	no	—
76	Dic₇₃	—	2	1	no	—
77	Dic₇₃	—	2	1	no	—
78	Dic₇₃	—	2	1	no	—
79	#7294	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	2	1	no	—
82	C₂×C₇₃	#1301	1	1	yes	characteristic
83	C₂×C₇₃	#1383	1	1	yes	characteristic
84	C₂×C₇₃	#1389	1	1	yes	characteristic
85	#2163	—	73	1	no	—
86	#1568	—	73	1	no	—
87	C₂×#334	—	73	1	no	—
88	C₇₃	#7294	1	1	yes	characteristic, complemented, Sylow
89	#432	—	146	1	no	—
90	C₂×C₄⋊C₈	—	146	1	no	—
91	C₂×C₄×C₈	—	73	1	no	—
92	C₄⋊C₁₆	—	146	1	no	—
93	C₂²⋊C₁₆	—	73	1	no	—
94	C₂²⋊C₁₆	#64408	1	1	yes	characteristic
95	#346	—	146	1	no	—
96	#335	—	146	1	no	—
97	#334	—	73	2	no	—
98	#334	—	73	2	no	—
99	#334	—	146	1	no	—
100	C₂²×C₈	—	292	1	no	—
101	C₂²×C₈	—	73	1	no	—
102	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
103	C₂×C₄²	—	73	1	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	146	1	no	—
106	C₄⋊C₈	—	292	1	no	—
107	C₄⋊C₈	—	146	2	no	—
108	C₄⋊C₈	—	146	2	no	—
109	C₂²⋊C₈	—	292	1	no	—
110	#97	—	292	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₄×C₈	—	73	1	no	—
114	C₄×C₈	—	73	1	no	—
115	C₂²×C₄	—	73	1	no	—
116	C₂²×C₄	—	73	1	no	—
117	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	292	2	no	—
120	C₂×C₈	—	292	2	no	—
121	C₂×C₈	—	584	1	no	—
122	C₂×C₈	—	584	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	146	2	no	—
126	C₂×C₈	C₄×#64408	1	1	yes	characteristic
127	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
128	C₄²	—	73	1	no	—
129	C₄²	—	73	1	no	—
130	C₄²	—	73	1	no	—
131	C₄²	—	73	1	no	—
132	C₁₆	—	4	1	no	—
133	C₁₆	—	292	1	no	—
134	C₂³	#14556659	1	1	yes	characteristic
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	292	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	73	1	no	—
142	C₂×C₄	—	146	1	no	—
143	C₂×C₄	—	146	1	no	—
144	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
145	C₂×C₄	#14556660	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	584	2	no	—
149	C₈	—	584	2	no	—
150	C₈	—	73	2	no	—
151	C₈	—	146	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	#17989461	1	1	yes	characteristic
155	C₂²	#17989511	1	1	yes	characteristic
156	C₂²	#17989512	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	146	1	no	—
159	C₄	—	146	1	no	—
160	C₄	—	146	1	no	—
161	C₄	—	146	1	no	—
162	C₄	#17989462	1	1	yes	characteristic
163	C₄	#17989468	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	#22624822	1	1	yes	characteristic
167	C₂	#22624996	1	1	yes	characteristic
168	C₂	#22625012	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625491

C₂

yes

maximal, characteristic

C₂×#17989449

C₂

yes

maximal, characteristic

#22611731

C₂

yes

maximal, characteristic

—

—

—

—

C₄

yes

—

#17989448

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988405

—

C₂²⋊#11924003

C₄

yes

characteristic

#17988362

—

C₂²⋊C₁₆×C₇₃

C₄

yes

—

—

—

—

—

—

Q₈

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

characteristic

C₂×#11924003

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

—

—

—

—

C₂²×Dic₇₃

SD₁₆

yes

characteristic

C₂²×Dic₇₃

SD₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

#11924003

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

—

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

#108

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

#7294

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

C₇₃

#7294

yes

characteristic, complemented, Sylow

#432

—

146

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

#64408

yes

characteristic

#346

—

146

—

#335

—

146

—

#334

—

#334

—

#334

—

146

—

100

C₂²×C₈

—

292

—

101

C₂²×C₈

—

102

C₂²×C₈

C₂×#64408

yes

characteristic

103

C₂×C₄²

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

146

—

106

C₄⋊C₈

—

292

—

107

C₄⋊C₈

—

146

—

108

C₄⋊C₈

—

146

—

109

C₂²⋊C₈

—

292

—

110

#97

—

292

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₄⋊#64408

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

292

—

120

C₂×C₈

—

292

—

121

C₂×C₈

—

584

—

122

C₂×C₈

—

584

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

146

—

126

C₂×C₈

C₄×#64408

yes

characteristic

127

C₂×C₈

C₂²⋊#64408

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₁₆

—

133

C₁₆

—

292

—

134

C₂³

#14556659

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

292

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

145

C₂×C₄

#14556660

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

584

—

149

C₈

—

584

—

150

C₈

—

151

C₈

—

146

—

152

C₂²

—

153

C₂²

—

154

C₂²

#17989461

yes

characteristic

155

C₂²

#17989511

yes

characteristic

156

C₂²

#17989512

yes

characteristic

157

C₄

—

158

C₄

—

146

—

159

C₄

—

146

—

160

C₄

—

146

—

161

C₄

—

146

—

162

C₄

#17989462

yes

characteristic

163

C₄

#17989468

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

#22624822

yes

characteristic

167

C₂

#22624996

yes

characteristic

168

C₂

#22625012

yes

characteristic

	subgroup	G/H
center	C₂²	#17989461
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂²⋊C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#106
cosocle	C₂×C₈×Dic₇₃	C₂²
hypercenter	C₂²⋊C₁₆	#64408
hypocenter	C₇₃	#7294
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#17989461

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂²⋊C₁₆×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₂×C₈×Dic₇₃

C₂²

C₂²⋊C₁₆

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
88. C₇₃	#7294	semidirect

H ≅ G/N

type

88. C₇₃

#7294

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22625491	2
C₂×#17989449	2
#22611731	2
#7294	73

subgroup

index

Subgroups — #34000656

Conjugacy classes of subgroups

Complements

Maximal subgroups